Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{r^2 - 36}{r + 6}$
Explanation: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{36} = 6$ So we can rewrite the expression as: $z = \dfrac{({r} + {6})({r} {-6})} {r + 6} $ We can divide the numerator and denominator by $(r + 6)$ on condition that $r \neq -6$ Therefore $z = r - 6; r \neq -6$